19 - Time series analysis
Francisco E. Fonturbel
21/February/2023
A very basic introduction to Time series analysis (TSA)
When we measure the same variable (whichever it is) many times in different times (e.g., daily temperature, monthly moth abundance, yearly inflation) we can build a times series dataset, allowing to conduct very interesting analyses. With TSA we can assess trends without the noise of temporal variation (e.g., seasons) in our data, and therefore making more robust inferences.
#Loading required packages
library(TSA)
library(forecast)
library(astsa)
#Loading work data
data<-read.table("data/19_bird_data.txt", header=T)
attach(data)
names(data)## [1] "year" "month" "aphspiC" "aphspiN" "carbarC" "carbarN" "elaalbC"
## [8] "elaalbN" "phrpatC" "phrpatN" "troaedC" "troaedN" "turfalC" "turfalN"
## [15] "zoncapC" "zoncapN"We will use now an unpublished dataset of bird abundance in two habitat types (notro and canelo forests) with monthly records for 12 year (yeah, a lot of work).
First, we need to convert each variable into a TSA object:
aphcan<-ts(aphspiC, start=c(2003, 1), end=c(2015, 12), frequency=12)
aphnot<-ts(aphspiN, start=c(2003, 1), end=c(2015, 12), frequency=12)
carcan<-ts(carbarC, start=c(2003, 1), end=c(2015, 12), frequency=12)
carnot<-ts(carbarN, start=c(2003, 1), end=c(2015, 12), frequency=12)
elacan<-ts(elaalbC, start=c(2003, 1), end=c(2015, 12), frequency=12)
elanot<-ts(elaalbN, start=c(2003, 1), end=c(2015, 12), frequency=12)
phrcan<-ts(phrpatC, start=c(2003, 1), end=c(2015, 12), frequency=12)
phrnot<-ts(phrpatN, start=c(2003, 1), end=c(2015, 12), frequency=12)
trocan<-ts(troaedC, start=c(2003, 1), end=c(2015, 12), frequency=12)
tronot<-ts(troaedN, start=c(2003, 1), end=c(2015, 12), frequency=12)
turcan<-ts(turfalC, start=c(2003, 1), end=c(2015, 12), frequency=12)
turnot<-ts(turfalN, start=c(2003, 1), end=c(2015, 12), frequency=12)
zoncan<-ts(zoncapC, start=c(2003, 1), end=c(2015, 12), frequency=12)
zonnot<-ts(zoncapN, start=c(2003, 1), end=c(2015, 12), frequency=12)Now that we have those object, we can plot each variable:
plot(aphcan,ylab='Rayadito abundance (Canelo)')
plot(aphnot,ylab='Rayadito abundance (Notro)')
plot(carcan,ylab='Jilguero abundance (Canelo)')
plot(carnot,ylab='Jilguero abundance (Notro)')
plot(elacan,ylab='Fiofio abundance (Canelo)')
plot(elanot,ylab='Fiofio abundance (Notro)')
plot(phrcan,ylab='Cometocino abundance (Canelo)')
plot(phrnot,ylab='Cometocino abundance (Notro)')
plot(trocan,ylab='Chercan abundance (Canelo)')
plot(tronot,ylab='Chercan abundance (Notro)')
plot(turcan,ylab='Zorzal abundance (Canelo)')
plot(turnot,ylab='Zorzal abundance (Notro)')
plot(zoncan,ylab='Chincol abundance (Canelo)')
plot(zonnot,ylab='Chincol abundance (Notro)')
Holy Molly! a lot of data! Let’s unify habitat data by species:
aphspi<-cbind(aphspiC,aphspiN)
carbar<-cbind(carbarC,carbarN)
elaalb<-cbind(elaalbC,elaalbN)
phrpat<-cbind(phrpatC,phrpatN)
troaed<-cbind(troaedC,troaedN)
turfal<-cbind(turfalC,turfalN)
zoncap<-cbind(zoncapC,zoncapN)And convert them in TSA objects:
rayadito<-ts(aphspi, start=c(2003, 1), end=c(2015, 12), frequency=12)
jilguero<-ts(carbar, start=c(2003, 1), end=c(2015, 12), frequency=12)
fiofio<-ts(elaalb, start=c(2003, 1), end=c(2015, 12), frequency=12)
cometocino<-ts(phrpat, start=c(2003, 1), end=c(2015, 12), frequency=12)
chercan<-ts(troaed, start=c(2003, 1), end=c(2015, 12), frequency=12)
zorzal<-ts(turfal, start=c(2003, 1), end=c(2015, 12), frequency=12)
chincol<-ts(zoncap, start=c(2003, 1), end=c(2015, 12), frequency=12)And plot them again:
plot(rayadito, main="Aphrastura spinicauda")
plot(jilguero, main="Carduelis barbata")
plot(fiofio, main="Elaenia albiceps")
plot(cometocino, main="Phrygilus patagonicus")
plot(chercan, main="Troglodytes aedon")
plot(zorzal, main="Turdus falcklandii")
plot(chincol, main="Zonotrichia capensis")
Autocorrelation functions
As we dealt with spatial autocorrelation on lesson 15, in TSA we will deal with temporal autocorrelation, that has the same principle (the shorter the measurement interval, the more similar a variable would be). In this case, we will estimate the autocorrelation function (ACF) for each variable:
acf(as.vector(aphcan),lag.max=36, main="ACF Rayadito Canelo")
acf(as.vector(aphnot),lag.max=36, main="ACF Rayadito Notro")
acf(as.vector(carcan),lag.max=36, main="ACF Jilguero Canelo")
acf(as.vector(carnot),lag.max=36, main="ACF Jilguero Notro")
acf(as.vector(elacan),lag.max=36, main="ACF Fiofio Canelo")
acf(as.vector(elanot),lag.max=36, main="ACF Fiofio Notro")
acf(as.vector(phrcan),lag.max=36, main="ACF Cometocino Canelo")
acf(as.vector(phrnot),lag.max=36, main="ACF Cometocino Notro")
acf(as.vector(trocan),lag.max=36, main="ACF Chercan Canelo")
acf(as.vector(tronot),lag.max=36, main="ACF Chercan Notro")
acf(as.vector(turcan),lag.max=36, main="ACF Zorzal Canelo")
acf(as.vector(turnot),lag.max=36, main="ACF Zorzal Notro")
acf(as.vector(zoncan),lag.max=36, main="ACF Chincol Canelo")
acf(as.vector(zonnot),lag.max=36, main="ACF Chincol Notro")
It’s common to find regular “waves” of autocorrelation, as they are given by seasonality. This analysis is useful to understand the temporal variation in any phenomenon that we are studying!
Decomposing time series data
This is my favorite part! We can decompose a time series data into four elements: the data itself, the seasonal variation, the trend, and the remainder (error). We also can plot the variation within each month across years, and annual variation across sampling years. We will use only two species (one resident and one migratory) as an example here:
#Rayadito-Canelo
decompose(aphcan)## $x
## Jan Feb Mar Apr May Jun
## 2003 7.0850202 2.5248275 3.9407314 5.1695616 2.6586317 8.8113490
## 2004 6.5971764 4.7471171 1.9413036 5.5741360 3.9841503 3.0158028
## 2005 2.3619220 4.0345014 2.7329374 6.2751660 7.4578334 6.9220120
## 2006 11.1505593 9.2762739 14.7637795 12.7519612 13.0383958 4.5471080
## 2007 0.9587728 0.0000000 7.8485688 4.7728900 1.8450184 10.0967040
## 2008 7.8895464 7.5322474 5.3806044 5.7974936 2.9285435 1.9599068
## 2009 1.9656664 7.6736964 8.2051282 5.6410256 7.6923077 10.7599055
## 2010 2.2624434 5.4240631 9.0905156 10.1382489 6.3572791 12.6047050
## 2011 5.6980057 5.8496636 4.0518639 4.1907245 2.8490028 3.3238367
## 2012 7.3902705 6.3398140 12.1794872 3.8461538 11.5384615 5.5555556
## 2013 1.3603468 4.3272413 5.7354926 3.6521342 3.8948393 2.7195027
## 2014 2.1680687 2.9749256 5.4439545 3.1730203 3.6505867 3.2796661
## 2015 4.7520048 5.5660692 6.6312997 6.0798308 3.1080031 0.8140008
## Jul Aug Sep Oct Nov Dec
## 2003 3.8639279 2.0917220 2.7951699 1.2621376 0.9802864 1.6876498
## 2004 1.7526333 2.6746550 2.2704568 0.4658052 0.6625969 1.2903464
## 2005 3.1655587 0.7723200 1.4732459 1.1393986 1.3000351 2.1180719
## 2006 1.4240957 0.4709872 0.0000000 0.9186111 1.4249074 0.4594321
## 2007 1.6092694 1.6686134 0.5574136 0.0000000 0.5554013 1.9180421
## 2008 3.6805300 1.4869151 1.4670644 2.2087407 1.9733986 1.6527590
## 2009 1.5061299 2.3076923 0.7820035 0.0000000 0.5128205 0.9473285
## 2010 1.3805798 1.3733748 1.4513788 2.1367521 2.9416091 2.4654832
## 2011 2.8490028 4.0064103 1.6436555 1.0957703 0.4748338 2.1367521
## 2012 1.9230769 1.9230769 1.2820513 0.4273504 1.7101618 2.5641026
## 2013 1.1257036 1.7094017 0.8249124 0.8165932 0.9380863 0.1797268
## 2014 3.8693701 0.7279786 1.2610340 1.1984775 2.3000256 0.2747253
## 2015 4.3215212 3.8850039 0.7631258 1.1094675 0.3750516 1.6662038
##
## $seasonal
## Jan Feb Mar Apr May Jun Jul
## 2003 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2004 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2005 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2006 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2007 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2008 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2009 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2010 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2011 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2012 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2013 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2014 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## 2015 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## Aug Sep Oct Nov Dec
## 2003 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2004 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2005 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2006 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2007 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2008 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2009 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2010 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2011 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2012 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2013 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2014 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
## 2015 -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
##
## $trend
## Jan Feb Mar Apr May Jun Jul Aug
## 2003 NA NA NA NA NA NA 3.552258 3.624526
## 2004 3.123744 3.060063 3.062488 3.007445 2.961027 2.931236 2.738213 2.532051
## 2005 3.300611 3.280219 3.167738 3.162587 3.217213 3.278262 3.678943 4.263544
## 2006 6.218831 6.133715 6.059774 5.989189 5.985193 5.921286 5.427518 4.616349
## 2007 2.526048 2.583665 2.656791 2.641741 2.567236 2.591782 2.941340 3.543966
## 2008 3.236059 3.314791 3.345122 3.475055 3.626170 3.674199 3.416317 3.175383
## 2009 4.443328 4.386927 4.392582 4.272007 4.119119 4.028868 4.011841 3.930472
## 2010 4.322538 4.278377 4.267337 4.384259 4.574490 4.738946 4.945351 5.106233
## 2011 3.203874 3.374768 3.492489 3.457127 3.310970 3.194491 3.251305 3.342239
## 2012 4.882767 4.757381 4.655509 4.612591 4.636212 4.705491 4.472050 4.136946
## 2013 2.593391 2.551264 2.523313 2.520484 2.504533 2.373014 2.307320 2.284629
## 2014 2.304707 2.378133 2.355413 2.389496 2.462155 2.522861 2.634483 2.850112
## 2015 3.067408 3.217790 3.328587 3.304132 3.220216 3.197987 NA NA
## Sep Oct Nov Dec
## 2003 3.633812 3.567360 3.639447 3.453196
## 2004 2.535344 2.597538 2.771485 3.078980
## 2005 4.983236 5.754388 6.256778 6.390347
## 2006 3.941704 3.321109 2.522257 2.287099
## 2007 3.754978 3.694838 3.782676 3.488790
## 2008 3.298965 3.410134 3.602105 4.167262
## 2009 3.873629 4.097904 4.229662 4.250903
## 2010 4.914022 4.456265 4.062273 3.529392
## 2011 3.701312 4.025606 4.373310 4.828359
## 2012 3.784589 3.508005 3.181437 2.744784
## 2013 2.216135 2.184024 2.153884 2.167047
## 2014 3.007549 3.178139 3.276648 3.151304
## 2015 NA NA NA NA
##
## $random
## Jan Feb Mar Apr May Jun
## 2003 NA NA NA NA NA NA
## 2004 2.51130102 -0.03316349 -4.53050871 0.15912731 -1.09137549 -1.80329796
## 2005 -1.90081966 -0.96593524 -3.84412404 0.70501541 2.12612178 1.75588556
## 2006 3.96959694 1.42234110 5.29468159 4.35520800 4.93870444 -3.26204269
## 2007 -2.52940609 -4.30388248 1.78245377 -0.27641509 -2.83671643 5.61705679
## 2008 3.69135601 2.49723837 -1.37384191 -0.08512568 -2.81212447 -3.60215725
## 2009 -3.43979277 1.56655123 0.40322234 -1.03854510 1.45869054 4.84317240
## 2010 -3.02222546 -0.57453159 1.41385423 3.34642559 -0.33170973 5.97789389
## 2011 1.53200081 0.75467770 -2.84994931 -1.67396587 -2.57646570 -1.75851878
## 2012 1.54537250 -0.13778517 4.11465449 -3.17400112 4.78775071 -1.03779986
## 2013 -2.19517504 0.05575948 -0.19714449 -1.27591379 -0.72419200 -1.54137614
## 2014 -1.09876890 -1.12342572 -0.32078185 -1.62403961 -0.92606718 -1.13105987
## 2015 0.72246583 0.62806101 -0.10661091 0.36813515 -2.22671126 -4.27185089
## Jul Aug Sep Oct Nov Dec
## 2003 1.56275073 0.29879627 1.46325719 0.35407572 -0.33718546 0.37019934
## 2004 0.26550119 1.97420411 2.03701243 0.52756522 0.21308760 0.34711192
## 2005 0.73769588 -1.65962318 -1.20809076 -1.95569090 -2.63476713 -2.13652910
## 2006 -2.75234189 -2.31376119 -1.63980429 0.25680055 1.22462603 0.30807867
## 2007 -0.08099007 -0.04375193 -0.89566461 -1.03553948 -0.90529976 0.56499768
## 2008 1.51529308 0.14313288 0.46999883 1.45790485 0.69326918 -0.37875687
## 2009 -1.25463056 0.20882082 -0.78972552 -1.43860576 -1.39486621 -1.16782832
## 2010 -2.31369075 -1.90125751 -1.16074418 0.33978517 1.20131093 1.07183644
## 2011 0.84877889 2.49577235 0.24424257 -0.27053776 -1.57650080 -0.55586122
## 2012 -1.29789256 -0.38226851 -0.20063828 -0.42135652 0.85070039 1.95506461
## 2013 0.06946400 1.25637357 0.91067702 1.29186703 1.10617758 0.14842546
## 2014 2.48596724 -0.29053248 0.55538478 0.67963706 1.34535284 -0.74083342
## 2015 NA NA NA NA NA NA
##
## $figure
## [1] 0.962131 1.720218 3.409324 2.407564 2.114498 1.887865 -1.251081
## [8] -1.831601 -2.301899 -2.659298 -2.321975 -2.135746
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"fit.aphcan<-stl(aphcan, "periodic")
plot(fit.aphcan, main="Rayadito-Canelo")
monthplot(aphcan, main="Rayadito-Canelo")
seasonplot(aphcan, main="Rayadito-Canelo")
##
#Rayadito-Notro
decompose(aphnot)## $x
## Jan Feb Mar Apr May Jun
## 2003 6.1050061 1.0229133 1.8743440 2.4238429 0.0000000 0.8841733
## 2004 1.2498334 1.3792089 3.0971578 2.5996309 1.1214911 2.2024168
## 2005 5.4695021 0.7735627 0.0000000 6.2743130 10.7953122 2.6809939
## 2006 1.2797477 1.2501719 5.7850636 7.6086130 2.9871382 1.4792899
## 2007 1.0668943 0.0000000 4.8076923 12.9152245 18.0045943 0.0000000
## 2008 0.7396450 1.5110610 2.9077563 3.2573054 4.9162597 7.2663388
## 2009 1.4827751 3.6641669 7.1794872 7.1794872 5.3846154 4.5969966
## 2010 2.5655279 2.6361575 0.0000000 2.1367521 6.1253355 1.6025641
## 2011 0.0000000 0.6759497 6.8332683 0.9438414 5.1943146 3.7986705
## 2012 1.5172205 0.9506736 17.9487179 4.7008547 7.7787381 9.8181508
## 2013 0.0000000 0.3406087 1.6764058 2.7720028 2.3310023 0.0000000
## 2014 0.3448434 0.2249213 2.9239766 2.5471218 1.4652015 0.8212428
## 2015 0.0000000 0.7692308 0.5656109 0.0000000 3.1185031 0.8867213
## Jul Aug Sep Oct Nov Dec
## 2003 2.3257166 0.7137046 0.0000000 0.9686825 0.8617867 4.5033415
## 2004 5.2151239 1.1249353 0.4814915 0.3789688 0.2117460 2.0143523
## 2005 1.5308075 1.5692182 0.8986341 1.0724873 0.4783179 2.3670720
## 2006 0.6700616 1.5068184 1.0559105 0.0000000 0.5290445 0.5262936
## 2007 0.7938398 3.3444816 0.8361204 1.1072317 0.8325008 0.9861933
## 2008 1.4791951 2.9600979 1.4400576 0.0000000 1.3433477 0.0000000
## 2009 3.0769231 0.5354275 0.0000000 0.7692308 2.0512821 0.0000000
## 2010 3.9083319 1.2922818 0.9496676 1.4245014 0.0000000 0.9496676
## 2011 1.4245014 3.5612536 0.9496676 0.0000000 0.0000000 1.9513718
## 2012 4.5639366 1.2800601 0.8547009 0.0000000 0.0000000 0.4273504
## 2013 1.3262599 0.7849294 0.0000000 0.2070051 0.7987858 2.3093719
## 2014 0.0000000 1.0256410 0.2616431 0.9841745 0.0000000 1.2457178
## 2015 0.8183306 0.0000000 0.0000000 0.2399347 0.4633920 0.2616431
##
## $seasonal
## Jan Feb Mar Apr May Jun
## 2003 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2004 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2005 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2006 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2007 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2008 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2009 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2010 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2011 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2012 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2013 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2014 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## 2015 -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## Jul Aug Sep Oct Nov Dec
## 2003 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2004 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2005 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2006 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2007 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2008 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2009 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2010 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2011 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2012 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2013 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2014 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
## 2015 -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
##
## $trend
## Jan Feb Mar Apr May Jun Jul
## 2003 NA NA NA NA NA NA 1.6046604
## 2004 1.8723062 2.0098328 2.0470295 2.0425203 1.9908638 1.8600709 1.9321826
## 2005 2.7981786 2.6631772 2.6990699 2.7453475 2.7853513 2.8111551 2.6512786
## 2006 2.3230157 2.2845513 2.2885045 2.2503707 2.2077973 2.1332118 2.0476439
## 2007 3.4287019 3.5104286 3.5778400 3.6148168 3.6735954 3.7054019 3.7109290
## 2008 2.4034509 2.4159914 2.4251395 2.4041689 2.3793195 2.3595134 2.3493858
## 2009 3.1257576 3.0913016 2.9302713 2.9023202 2.9638687 2.9933660 3.0384807
## 2010 1.8262421 1.8924197 1.9635248 2.0303972 1.9722301 1.9263295 1.8590020
## 2011 2.0607150 2.0517625 2.1463030 2.0869488 2.0275946 2.0693322 2.1742874
## 2012 4.3475723 4.3833324 4.2843257 4.2803687 4.2803687 4.2168679 4.0901494
## 2013 1.0522691 0.8967355 0.8404925 0.8135052 0.8554131 0.9671134 1.0598994
## 2014 1.0908775 1.0456463 1.0665777 1.1098616 1.1089609 1.0313592 0.9726718
## 2015 0.7722007 0.7635627 0.7099259 0.6680141 0.6563121 0.6346170 NA
## Aug Sep Oct Nov Dec
## 2003 1.4172072 1.4830034 1.5412785 1.5953318 1.6969874
## 2004 2.0827668 1.9284833 1.9525469 2.5087345 2.9317511
## 2005 2.4965643 2.7574673 3.0541074 2.7843627 2.4089511
## 2006 1.9866845 1.8938702 2.0742552 2.9210913 3.4851816
## 2007 3.7602545 3.7440514 3.2624741 2.3147135 2.0721304
## 2008 2.4700623 2.7377638 3.0791769 3.2621159 3.1704081
## 2009 3.0407616 2.6987826 2.1895234 2.0102727 1.9163681
## 2010 1.6704296 1.8734738 2.1084887 2.0199916 2.0727035
## 2011 2.2489518 2.7235423 3.3432283 3.6074548 3.9659508
## 2012 4.0015126 3.2980802 2.5396983 2.2323405 1.5962619
## 2013 1.0694476 1.1166094 1.1592215 1.1137764 1.1119198
## 2014 0.9809829 0.9053972 0.7010019 0.6637594 0.7353753
## 2015 NA NA NA NA NA
##
## $random
## Jan Feb Mar Apr May Jun
## 2003 NA NA NA NA NA NA
## 2004 0.33492561 0.44741879 -1.17009245 -1.59980805 -4.38748498 -0.35278447
## 2005 3.62872190 -0.81157182 -4.91929071 1.37204683 4.49184870 -0.82529151
## 2006 -0.08586951 0.04366329 1.27633834 3.20132362 -2.73877138 -1.34905224
## 2007 -1.40440920 -2.43238596 -0.99036848 7.14348907 10.81288657 -4.40053226
## 2008 -0.70640753 0.17311221 -1.73760398 -1.30378211 -0.98117210 4.21169501
## 2009 -0.68558401 1.65090795 2.02899513 2.12024838 -1.09736561 0.90850032
## 2010 1.69668423 1.82178051 -4.18374555 -2.05056371 0.63499314 -1.01889570
## 2011 -1.10331652 -0.29777013 2.46674448 -3.30002598 -0.35139225 1.03420790
## 2012 -1.87295340 -2.35461607 11.44417149 -1.73643268 -0.01974291 4.90615258
## 2013 -0.09487066 0.52191595 -1.38430746 -0.19842102 -2.04252306 -1.66224375
## 2014 0.21136444 0.25731768 -0.36282188 -0.71965846 -3.16187171 -0.90524675
## 2015 0.18519779 1.08371073 -2.36453578 -2.82493273 -1.05592127 -0.44302598
## Jul Aug Sep Oct Nov Dec
## 2003 0.82766927 -0.06764767 0.14500616 1.11042085 0.93567293 3.63858240
## 2004 3.38955436 -0.32197656 0.18101771 0.10943875 -0.62777044 -0.08517049
## 2005 -1.01385807 -0.29149107 -0.23082363 -0.29860331 -0.63682676 0.79034923
## 2006 -1.27096914 0.15598886 0.79004988 -0.39123832 -0.72282870 -2.12665962
## 2007 -2.81047616 0.22008206 -1.27992140 -0.47222553 0.18700539 -0.25370874
## 2008 -0.76357761 1.12589056 0.33030336 -1.39616000 -0.24955010 -2.33817982
## 2009 0.14505547 -1.86947923 -1.07077304 0.26272426 1.71022738 -1.08413973
## 2010 2.15594295 0.25770717 0.70420337 0.99902956 -0.35077350 -0.29080752
## 2011 -0.64317293 1.94815676 -0.14586514 -1.66021144 -1.93823675 -1.18235068
## 2012 0.58040022 -2.08559751 -0.81536974 -0.85668148 -0.56312245 -0.33668316
## 2013 0.37297356 0.35133671 0.51140016 0.73080041 1.35422750 2.02968035
## 2014 -0.86605878 0.68051307 0.98425545 1.96618939 1.00545864 1.34257090
## 2015 NA NA NA NA NA NA
##
## $figure
## [1] -0.9573984 -1.0780427 2.2202208 2.1569186 3.5181123 0.6951303
## [7] -0.1066131 -0.6358550 -1.6280096 -1.6830169 -1.6692181 -0.8322283
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"fit.aphnot<-stl(aphnot, "periodic")
plot(fit.aphnot, main="Rayadito-Notro")
monthplot(aphnot, main="Rayadito-Notro")
seasonplot(aphnot, main="Rayadito-Notro")
##
#Chincol-Canelo
decompose(zoncan)## $x
## Jan Feb Mar Apr May Jun
## 2003 1.01214575 0.84160916 0.00000000 0.00000000 0.00000000 0.00000000
## 2004 1.64929410 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2005 0.23619220 0.21234218 0.00000000 0.00000000 0.00000000 0.00000000
## 2006 5.35226847 0.88345466 0.00000000 0.00000000 0.00000000 0.00000000
## 2007 3.35570470 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2008 1.47928994 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2009 0.00000000 0.45139390 0.00000000 0.00000000 0.00000000 0.00000000
## 2010 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2011 2.37416904 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2012 1.92789667 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2013 0.00000000 0.50908721 0.00000000 0.00000000 0.00000000 0.00000000
## 2014 0.08672275 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2015 4.45500446 0.55660692 0.00000000 0.00000000 0.00000000 0.00000000
## Jul Aug Sep Oct Nov Dec
## 2003 0.00000000 0.00000000 0.00000000 0.84142504 0.00000000 0.84382489
## 2004 0.00000000 0.00000000 0.56761420 0.69870786 0.66259685 0.36867039
## 2005 0.00000000 0.00000000 1.47324586 0.22787972 0.21667252 0.00000000
## 2006 0.00000000 0.00000000 1.70523336 0.45930553 0.94993825 0.45943214
## 2007 0.00000000 0.00000000 0.55741360 0.50549643 0.55540128 0.00000000
## 2008 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.55091967
## 2009 0.00000000 0.00000000 0.78200349 0.00000000 0.00000000 0.00000000
## 2010 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## 2011 0.00000000 0.00000000 0.00000000 0.54788516 0.00000000 0.00000000
## 2012 0.00000000 0.00000000 1.28205128 0.42735043 0.42754044 0.00000000
## 2013 0.00000000 0.00000000 0.00000000 0.61244488 0.00000000 0.00000000
## 2014 0.00000000 0.00000000 0.00000000 0.00000000 0.25555839 0.54945055
## 2015 0.00000000 0.00000000 0.00000000 0.73964497 0.18752579 0.00000000
##
## $seasonal
## Jan Feb Mar Apr May Jun
## 2003 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2004 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2005 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2006 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2007 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2008 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2009 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2010 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2011 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2012 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2013 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2014 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## 2015 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## Jul Aug Sep Oct Nov Dec
## 2003 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2004 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2005 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2006 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2007 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2008 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2009 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2010 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2011 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2012 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2013 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2014 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
## 2015 -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
##
## $trend
## Jan Feb Mar Apr May Jun
## 2003 NA NA NA NA NA NA
## 2004 0.27787867 0.27787867 0.30152926 0.31923330 0.34089496 0.34870505
## 2005 0.22884364 0.22884364 0.26657829 0.28469511 0.24649709 0.21255564
## 2006 0.67946010 0.67946010 0.68912625 0.70843514 0.74863062 0.79832636
## 2007 0.57746783 0.57746783 0.52964201 0.48374081 0.46922639 0.43364434
## 2008 0.25813344 0.25813344 0.23490787 0.19061995 0.14641588 0.14622915
## 2009 0.08352613 0.08352613 0.11610961 0.14869309 0.14869309 0.12573810
## 2010 0.06516696 0.06516696 0.03258348 0.00000000 0.00000000 0.00000000
## 2011 0.19784742 0.19784742 0.19784742 0.22067597 0.24350452 0.24350452
## 2012 0.20631515 0.20631515 0.25973396 0.30813048 0.32092238 0.33873657
## 2013 0.22050245 0.22050245 0.16708364 0.12137711 0.11127519 0.09346101
## 2014 0.05826397 0.05826397 0.05826397 0.03274543 0.01787516 0.05141720
## 2015 0.48471836 0.48471836 0.48471836 0.51553690 0.54352075 0.51779228
## Jul Aug Sep Oct Nov Dec
## 2003 0.32146492 0.31294572 0.27787867 0.27787867 0.27787867 0.27787867
## 2004 0.27002770 0.21999605 0.22884364 0.22884364 0.22884364 0.22884364
## 2005 0.41036422 0.65149708 0.67946010 0.67946010 0.67946010 0.67946010
## 2006 0.73427921 0.61427844 0.57746783 0.57746783 0.57746783 0.57746783
## 2007 0.33631739 0.25813344 0.25813344 0.25813344 0.25813344 0.25813344
## 2008 0.10754705 0.06471805 0.08352613 0.08352613 0.08352613 0.08352613
## 2009 0.10278312 0.08397504 0.06516696 0.06516696 0.06516696 0.06516696
## 2010 0.09892371 0.19784742 0.19784742 0.19784742 0.19784742 0.19784742
## 2011 0.22490983 0.20631515 0.20631515 0.20631515 0.20631515 0.20631515
## 2012 0.25840754 0.19929048 0.22050245 0.22050245 0.22050245 0.22050245
## 2013 0.09707446 0.07947594 0.05826397 0.05826397 0.05826397 0.05826397
## 2014 0.25632271 0.46152641 0.48471836 0.48471836 0.48471836 0.48471836
## 2015 NA NA NA NA NA NA
##
## $random
## Jan Feb Mar Apr May
## 2003 NA NA NA NA NA
## 2004 -0.0924753203 -0.2164646348 -0.0223748205 -0.0404322662 -0.0617961909
## 2005 -1.4565421894 0.0449125711 0.0125761481 -0.0058940679 0.0326016800
## 2006 3.2089176151 0.2654085936 -0.4099718074 -0.4296340987 -0.4695318518
## 2007 1.3143461126 -0.5160537979 -0.2504875684 -0.2049397690 -0.1901276210
## 2008 -0.2427342498 -0.1967194033 0.0442465695 0.0881810845 0.1326828839
## 2009 -1.5474168844 0.4292818071 0.1630448303 0.1301079485 0.1304056766
## 2010 -1.5290577110 -0.0037529235 0.2465709614 0.2788010373 0.2790987655
## 2011 0.7124308676 -0.1364333859 0.0813070202 0.0581250688 0.0355942485
## 2012 0.2576907594 -0.1449011181 0.0194204846 -0.0293294412 -0.0418236176
## 2013 -1.6843932000 0.3499987945 0.1120707970 0.1574239285 0.1678235730
## 2014 -1.4354319752 0.0031500653 0.2208904714 0.2460556050 0.2612236034
## 2015 2.5063953415 0.1333025980 -0.2055639199 -0.2367358634 -0.2644219835
## Jun Jul Aug Sep Oct
## 2003 NA -0.0522856654 -0.0328016858 -0.5293543784 0.4826595550
## 2004 -0.0718851051 -0.0008484524 0.0601479820 0.0872948546 0.3889774029
## 2005 0.0642643105 -0.1411849649 -0.3713530500 0.5423100441 -0.5324671986
## 2006 -0.5215064131 -0.4650999590 -0.3341344116 0.8762898196 -0.1990491151
## 2007 -0.1568243907 -0.0671381337 0.0220105939 0.0478044541 0.1664761805
## 2008 0.1305908012 0.1616321987 0.2154259796 -0.3350018404 -0.1644129441
## 2009 0.1510818465 0.1663961356 0.1961689943 0.4653608259 -0.1460537708
## 2010 0.2768199491 0.1702555419 0.0822966113 -0.4493231294 -0.2787342331
## 2011 0.0333154321 0.0442694173 0.0738288791 -0.4577908617 0.2606831976
## 2012 -0.0619166191 0.0107717114 0.0808535516 0.8100731259 0.1259611672
## 2013 0.1833589419 0.1721047969 0.2006680955 -0.3097396783 0.4732940981
## 2014 0.2254027478 0.0128565399 -0.1813823737 -0.7361940696 -0.5656051733
## 2015 -0.2409723349 NA NA NA NA
## Nov Dec
## 2003 -0.2543665402 0.6140758571
## 2004 0.4572653387 0.1879563870
## 2005 -0.4392754558 -0.6313304645
## 2006 0.3959825507 -0.0699060530
## 2007 0.3207799683 -0.2100038004
## 2008 -0.0600140023 0.5155231750
## 2009 -0.0416548289 -0.0170373207
## 2010 -0.1743352913 -0.1497177830
## 2011 -0.1828030235 -0.1581855153
## 2012 0.2305501271 -0.1723728097
## 2013 -0.0347518401 -0.0101343319
## 2014 -0.2056478364 0.1128618258
## 2015 NA NA
##
## $figure
## [1] 1.46389075 -0.06141403 -0.27915444 -0.27880104 -0.27909877 -0.27681995
## [7] -0.26917925 -0.28014403 0.25147571 0.08088681 -0.02351213 -0.04812964
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"fit.zoncan<-stl(zoncan, "periodic")
plot(fit.zoncan, main="Chincol-Canelo")
monthplot(zoncan, main="Chincol-Canelo")
seasonplot(zoncan, main="Chincol-Canelo")
##
#Chincol-Notro
decompose(zonnot)## $x
## Jan Feb Mar Apr May Jun Jul
## 2003 0.0000000 0.0000000 0.6247813 0.0000000 0.0000000 0.0000000 0.0000000
## 2004 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2005 1.0939004 0.5157085 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2006 0.2559495 2.1878008 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2007 1.6003414 0.5341880 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2008 2.9585799 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2009 0.9885167 0.7328334 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2010 0.0000000 0.4393596 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2011 2.7870680 0.0000000 0.3253937 0.0000000 0.0000000 0.0000000 0.0000000
## 2012 2.5287008 1.0864841 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2013 0.1451817 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2014 0.4597913 0.6747638 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2015 0.9230769 1.1538462 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## Aug Sep Oct Nov Dec
## 2003 0.0000000 1.0016026 0.4843412 2.1544666 0.6004455
## 2004 0.0000000 2.4074573 0.0000000 0.2117460 1.0071761
## 2005 0.0000000 0.0000000 0.2144975 0.7174768 0.0000000
## 2006 0.0000000 0.0000000 0.5152249 0.0000000 1.0525873
## 2007 0.0000000 0.0000000 1.6608475 0.0000000 1.4792899
## 2008 0.0000000 0.0000000 2.5625081 0.0000000 0.0000000
## 2009 0.0000000 1.0311404 0.0000000 1.5384615 0.4748338
## 2010 0.3230705 2.3741690 2.8490028 0.0000000 0.9496676
## 2011 0.0000000 2.3741690 2.8490028 1.8844458 1.1708231
## 2012 0.0000000 2.1367521 0.0000000 0.6410256 0.4273504
## 2013 0.0000000 2.1770682 0.8280202 0.3993929 1.0497145
## 2014 0.0000000 1.8315018 0.9841745 0.4180602 0.6228589
## 2015 0.0000000 0.2590003 0.4798695 0.0000000 0.2616431
##
## $seasonal
## Jan Feb Mar Apr May Jun
## 2003 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2004 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2005 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2006 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2007 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2008 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2009 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2010 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2011 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2012 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2013 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2014 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## 2015 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## Jul Aug Sep Oct Nov Dec
## 2003 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2004 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2005 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2006 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2007 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2008 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2009 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2010 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2011 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2012 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2013 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2014 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
## 2015 -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
##
## $trend
## Jan Feb Mar Apr May Jun Jul
## 2003 NA NA NA NA NA NA 0.4054698
## 2004 0.3534047 0.3534047 0.4119819 0.4503783 0.3492508 0.2852512 0.3477775
## 2005 0.4363324 0.4363324 0.3360216 0.2446483 0.2746578 0.2537643 0.1768840
## 2006 0.2813104 0.2813104 0.2813104 0.2938407 0.2764761 0.2904391 0.3903132
## 2007 0.3085285 0.3085285 0.3085285 0.3562627 0.4039970 0.4217763 0.4961488
## 2008 0.5082264 0.5082264 0.5082264 0.5457956 0.5833648 0.5217277 0.3780047
## 2009 0.3569882 0.3569882 0.3999524 0.3361454 0.2934768 0.3773641 0.3559606
## 2010 0.2903163 0.3037776 0.3731983 0.5478663 0.6024722 0.5581544 0.6940670
## 2011 0.8006976 0.7872364 0.7737751 0.7737751 0.8522937 0.9400271 0.9384766
## 2012 0.9911355 0.9911355 0.9812431 0.8526423 0.6821247 0.5993375 0.4690461
## 2013 0.2791925 0.2791925 0.2808723 0.3170530 0.3414858 0.3573496 0.3963902
## 2014 0.4657292 0.4657292 0.4513306 0.4434385 0.4507227 0.4337149 0.4352328
## 2015 0.4944599 0.4944599 0.4289390 0.3424054 0.3039735 0.2715037 NA
## Aug Sep Oct Nov Dec
## 2003 0.4054698 0.3794372 0.3534047 0.3534047 0.3534047
## 2004 0.4148445 0.4363324 0.4363324 0.4363324 0.4363324
## 2005 0.2116399 0.2813104 0.2813104 0.2813104 0.2813104
## 2006 0.3774290 0.3085285 0.3085285 0.3085285 0.3085285
## 2007 0.5304843 0.5082264 0.5082264 0.5082264 0.5082264
## 2008 0.3264535 0.3569882 0.3569882 0.3569882 0.3569882
## 2009 0.3025444 0.2903163 0.2903163 0.2903163 0.2903163
## 2010 0.7918881 0.7871396 0.8006976 0.8006976 0.8006976
## 2011 0.9729814 1.0046935 0.9911355 0.9911355 0.9911355
## 2012 0.3244627 0.2791925 0.2791925 0.2791925 0.2791925
## 2013 0.4376141 0.4657292 0.4657292 0.4657292 0.4657292
## 2014 0.4744981 0.4944599 0.4944599 0.4944599 0.4944599
## 2015 NA NA NA NA NA
##
## $random
## Jan Feb Mar Apr May
## 2003 NA NA NA NA NA
## 2004 -1.030793156 -0.496116307 0.026027156 0.012152895 0.105784161
## 2005 -0.019820441 -0.063335523 0.101987465 0.217882930 0.180377105
## 2006 -0.702749328 1.763778798 0.156698720 0.168690546 0.178558788
## 2007 0.614424449 0.082947926 0.129480637 0.106268497 0.051037905
## 2008 1.772964943 -0.650938089 -0.070217344 -0.083264398 -0.128329902
## 2009 -0.045859945 0.233133551 0.038056733 0.126385856 0.161558149
## 2010 -0.967704772 -0.007129603 0.064810759 -0.085335082 -0.147437284
## 2011 1.308981871 -0.929948014 -0.010372274 -0.311243863 -0.397258751
## 2012 0.860176792 -0.047362976 -0.543233997 -0.390111037 -0.227089726
## 2013 -0.811399275 -0.421904134 0.157136774 0.145478232 0.113549106
## 2014 -0.683326478 0.066322949 -0.013321540 0.019092768 0.004312223
## 2015 -0.248771444 0.516674637 0.009070127 0.120125871 0.151061440
## Jun Jul Aug Sep Oct
## 2003 NA 0.055354762 0.035643762 -0.185783267 -0.480327994
## 2004 0.161126563 0.113047067 0.026269030 1.163176368 -1.047596937
## 2005 0.192613476 0.283940564 0.229473669 -1.089258994 -0.678077505
## 2006 0.155938667 0.070511333 0.063684539 -1.116477077 -0.404568146
## 2007 0.024601443 -0.035324309 -0.089370745 -1.316175058 0.541356526
## 2008 -0.075350005 0.082819836 0.114660077 -1.164936796 1.594255354
## 2009 0.069013659 0.104863910 0.138569182 -0.067124451 -0.901580857
## 2010 -0.111776646 -0.233242425 -0.027704147 0.779080861 1.437040632
## 2011 -0.493649321 -0.477652034 -0.531867905 0.561526887 1.246602802
## 2012 -0.152959710 -0.008221593 0.116650873 1.049611034 -0.890457068
## 2013 0.089028117 0.064434346 0.003499455 0.903390362 -0.248973614
## 2014 0.012662884 0.025591759 -0.033384574 0.529093346 -0.121549977
## 2015 0.174874089 NA NA NA NA
## Nov Dec
## 2003 1.605009412 -0.021484340
## 2004 -0.420638951 0.302318563
## 2005 0.240113832 -0.549835589
## 2006 -0.504581029 0.475533587
## 2007 -0.704279010 0.702538287
## 2008 -0.553040748 -0.625513391
## 2009 1.052092694 -0.084007679
## 2010 -0.996750204 -0.119555231
## 2011 0.697257750 -0.088837588
## 2012 0.165780586 -0.120367272
## 2013 -0.262388882 0.315460030
## 2014 -0.272452237 -0.140126159
## 2015 NA NA
##
## $figure
## [1] 0.6773885 0.1427116 -0.4380091 -0.4625312 -0.4550349 -0.4463777
## [7] -0.4608245 -0.4411135 0.8079486 0.6112646 0.1960526 0.2685252
##
## $type
## [1] "additive"
##
## attr(,"class")
## [1] "decomposed.ts"fit.zonnot<-stl(zonnot, "periodic")
plot(fit.zonnot, main="Chincol-Notro")
monthplot(zonnot, main="Chincol-Notro")
seasonplot(zonnot, main="Chincol-Notro")
Very useful information indeed, with a bunch of applications for population ecology, conservation biology, wildlife management, etc.
ARIMA models
Autoregressive integrated moving average (aka ARIMA) are intended to be used with time series data as they account for seasonality and temporal autocorrelation. Below we will compare one species in two habitats as an example using ARIMA models of first order. ARIMA could be decomposed in AR-I-MA and 0 or 1 indicate if each component (autoregressive, integrated, or moving average) is present or not in the model. A detailed explanation of these models is beyond the scope of this lesson, but there are many good books on this subject.
##Rayadito-Canelo
m1.aphcan=arima(aphcan,order=c(1,1,1),seasonal=list(order=c(0,1,1), period=12))
m1.aphcan##
## Call:
## arima(x = aphcan, order = c(1, 1, 1), seasonal = list(order = c(0, 1, 1), period = 12))
##
## Coefficients:
## ar1 ma1 sma1
## 0.3233 -0.9927 -0.9990
## s.e. 0.0851 0.1312 0.3394
##
## sigma^2 estimated as 5.162: log likelihood = -338.61, aic = 683.22#Diagnosing ARIMA model
tsdiag(m1.aphcan)
Box.test(m1.aphcan$residuals,lag=1)##
## Box-Pierce test
##
## data: m1.aphcan$residuals
## X-squared = 0.69424, df = 1, p-value = 0.4047#Estimating ma1 significance
(1-pnorm(abs(m1.aphcan$coef)/sqrt(diag(m1.aphcan$var.coef))))*2## ar1 ma1 sma1
## 1.449306e-04 3.907985e-14 3.245826e-03# Residuals
plot(window(rstandard(m1.aphcan),start=c(2003,2)), ylab='Standardized Residuals',type='o')
abline(h=0)
# ACF
acf(as.vector(window(rstandard(m1.aphcan),start=c(2003,2))), lag.max=36)
hist(window(rstandard(m1.aphcan),start=c(2003,2)), xlab='Standardized Residuals')
##Rayadito-Notro
m1.aphnot=arima(aphnot,order=c(1,1,1),seasonal=list(order=c(0,1,1), period=12))
m1.aphnot##
## Call:
## arima(x = aphnot, order = c(1, 1, 1), seasonal = list(order = c(0, 1, 1), period = 12))
##
## Coefficients:
## ar1 ma1 sma1
## 0.2302 -0.9569 -0.9093
## s.e. 0.0886 0.0354 0.1615
##
## sigma^2 estimated as 6.609: log likelihood = -349.66, aic = 705.31#Diagnosing ARIMA model
tsdiag(m1.aphnot)
Box.test(m1.aphnot$residuals,lag=1)##
## Box-Pierce test
##
## data: m1.aphnot$residuals
## X-squared = 0.11772, df = 1, p-value = 0.7315#Estimating ma1 significance
(1-pnorm(abs(m1.aphnot$coef)/sqrt(diag(m1.aphnot$var.coef))))*2## ar1 ma1 sma1
## 9.335160e-03 0.000000e+00 1.778289e-08# Residuals
plot(window(rstandard(m1.aphnot),start=c(2003,2)), ylab='Standardized Residuals',type='o')
abline(h=0)
# ACF
acf(as.vector(window(rstandard(m1.aphnot),start=c(2003,2))), lag.max=36)
hist(window(rstandard(m1.aphnot),start=c(2003,2)), xlab='Standardized Residuals')
Final thoughts
aaa
Session
sessionInfo()## R version 4.1.0 (2021-05-18)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Big Sur 10.16
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] astsa_1.14 forecast_8.16 TSA_1.3
##
## loaded via a namespace (and not attached):
## [1] zoo_1.8-9 locfit_1.5-9.5 tidyselect_1.1.2 xfun_0.30
## [5] bslib_0.3.1 urca_1.3-0 purrr_0.3.4 splines_4.1.0
## [9] lattice_0.20-45 colorspace_2.0-3 vctrs_0.4.0 generics_0.1.2
## [13] htmltools_0.5.2 yaml_2.3.5 mgcv_1.8-40 utf8_1.2.2
## [17] rlang_1.0.2 jquerylib_0.1.4 pillar_1.7.0 glue_1.6.2
## [21] DBI_1.1.2 TTR_0.24.3 lifecycle_1.0.1 quantmod_0.4.18
## [25] stringr_1.4.0 timeDate_3043.102 munsell_0.5.0 gtable_0.3.0
## [29] leaps_3.1 evaluate_0.15 tseries_0.10-50 knitr_1.38
## [33] fastmap_1.1.0 lmtest_0.9-40 curl_4.3.2 parallel_4.1.0
## [37] fansi_1.0.3 highr_0.9 xts_0.12.1 Rcpp_1.0.8.3
## [41] scales_1.2.0 jsonlite_1.8.0 fracdiff_1.5-1 ggplot2_3.3.5
## [45] digest_0.6.29 stringi_1.7.6 dplyr_1.0.8 grid_4.1.0
## [49] quadprog_1.5-8 cli_3.2.0 tools_4.1.0 magrittr_2.0.3
## [53] sass_0.4.1 tibble_3.1.6 crayon_1.5.1 pkgconfig_2.0.3
## [57] ellipsis_0.3.2 Matrix_1.4-1 assertthat_0.2.1 rmarkdown_2.13
## [61] rstudioapi_0.13 R6_2.5.1 nnet_7.3-17 nlme_3.1-157
## [65] compiler_4.1.0Licensing Creative Commons License
Licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
2022 - Francisco E. Fontúrbel